Two beams are typically compared using their first and second moments, followed by qualitative examination of the beam phase space. For beams at high intensity, in the presence of collective instabilities, or in the presence of highly nonlinear transport, the details of the distribution (including higher-order moments and nonlinear correlations) become increasingly important. Such systems may exhibit filamentation of
the beam phase space, developing structure on finer and finer scales, in some cases relaxing to a quasi-stationary state. It is then valuable to have quantitative particle-based diagnostics that can characterize the nonlinear dynamical processes of regular or chaotic mixing and collisionless relaxation [1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
